| L(s) = 1 | − 4-s − 2·7-s − 4·13-s − 3·16-s − 8·19-s + 2·28-s − 8·31-s − 4·37-s + 8·43-s + 3·49-s + 4·52-s − 20·61-s + 7·64-s + 8·67-s − 28·73-s + 8·76-s + 16·79-s + 8·91-s − 28·97-s + 8·103-s + 4·109-s + 6·112-s − 10·121-s + 8·124-s + 127-s + 131-s + 16·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s − 1.10·13-s − 3/4·16-s − 1.83·19-s + 0.377·28-s − 1.43·31-s − 0.657·37-s + 1.21·43-s + 3/7·49-s + 0.554·52-s − 2.56·61-s + 7/8·64-s + 0.977·67-s − 3.27·73-s + 0.917·76-s + 1.80·79-s + 0.838·91-s − 2.84·97-s + 0.788·103-s + 0.383·109-s + 0.566·112-s − 0.909·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.152802080925025956854116387692, −9.041739330556870645526973791472, −8.486047191146135179383478045838, −8.138830421689102299251651732368, −7.48318071343520741736810069807, −7.26534813260720064108662301111, −6.87853029702903180920342868933, −6.32411000044886643798810545930, −6.05420919147302561108896002088, −5.55949584772863814681134226966, −4.91222842779235652798525196373, −4.72453124273459494740236995495, −3.97601938448959508043292558696, −3.95821704618757537757290215994, −3.11353433074374222256109561769, −2.55220020995291457056915011569, −2.17644355062192818414262405510, −1.41141467297432757952072339356, 0, 0,
1.41141467297432757952072339356, 2.17644355062192818414262405510, 2.55220020995291457056915011569, 3.11353433074374222256109561769, 3.95821704618757537757290215994, 3.97601938448959508043292558696, 4.72453124273459494740236995495, 4.91222842779235652798525196373, 5.55949584772863814681134226966, 6.05420919147302561108896002088, 6.32411000044886643798810545930, 6.87853029702903180920342868933, 7.26534813260720064108662301111, 7.48318071343520741736810069807, 8.138830421689102299251651732368, 8.486047191146135179383478045838, 9.041739330556870645526973791472, 9.152802080925025956854116387692
